Why Use a Fuzzy Partition in F-Transform?
Abstract
:1. Formulation of the Problem
1.1. F-Transform: A Brief Reminder
1.2. The General Idea behind F-Transform Is Very Reasonable
1.3. However, Why a Fuzzy Partition?
1.3.1. Mathematical Comment
1.3.2. Application-Related Comment
1.4. It Is Desirable to Explain the Efficiency of a Fuzzy Partition Requirement
1.5. What We Do in this Paper
1.6. The Structure of this Paper
Comment
2. Main Idea
2.1. What if We Can Make Exact Measurements of Instantaneous Values?
- we reconstruct the values , , …, with perfect accuracy (0 measurement error), while
- the values corresponding to all other moments of time t are reconstructed with no accuracy at all (the only bound on measurement error is infinity).
- we know the values with finite accuracy, but
- for all other moments of time t, we know nothing (i.e., the only bound of measurement error is infinity).
2.2. Main Idea
Comment
3. Case of Probabilistic Uncertainty
3.1. Description of the Case
- that each measurement error is normally distributed with 0 mean and known standard deviation , and
- that measurement errors and corresponding to different measurements are independent.
3.2. How Accurately Can We Estimate Based on Each Measurement
3.3. How Accurately Can We Estimate Based on All The Measurements
3.4. Discussion
- in the fuzzy partition requirement, we demand that the sum of the functions be constant, but
- here, we have the sum of the squares.
4. How Uncertainties Can Be Combined in Different Approaches
4.1. Towards a General Formulation of the Problem
4.2. Commutativity
4.3. Associativity
- we can first combine the first and the second ones, and then combine the result with the third one,
- or we can first combine the second and the third ones, and then combine the result with the first one.
4.4. Monotonicity
4.5. Non-Degenerate Case
4.6. Scale-Invariance
4.7. Discussion
- for all a and b, we have (commutativity);
- for all a, b, and c, we have (associativity);
- for all a and b, we have (first monotonicity requirement);
- for all a, b, , and , if and , then (second monotonicity requirement);
- if and , then (non-degeneracy); and
- for all a, b, and , we have (scale-invariance).
Comment
Comment
4.8. Discussion
5. Which Functions Should We Choose: General Uncertainty Situation and Case of Fuzzy Uncertainty
5.1. Analysis of the Problem
5.2. General Conclusion
5.3. Which Value Should We Use in the Case of Fuzzy Uncertainty
6. Conclusions and Future Work
6.1. Conclusions
6.2. Possible Directions of Future Research
- to different imprecise probability situations, and
- to situations when different functions correspond to different types of uncertainty.
Author Contributions
Funding
Acknowledgments
Conflicts of Interest
References
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Kreinovich, V.; Kosheleva, O.; Sriboonchitta, S. Why Use a Fuzzy Partition in F-Transform? Axioms 2019, 8, 94. https://doi.org/10.3390/axioms8030094
Kreinovich V, Kosheleva O, Sriboonchitta S. Why Use a Fuzzy Partition in F-Transform? Axioms. 2019; 8(3):94. https://doi.org/10.3390/axioms8030094
Chicago/Turabian StyleKreinovich, Vladik, Olga Kosheleva, and Songsak Sriboonchitta. 2019. "Why Use a Fuzzy Partition in F-Transform?" Axioms 8, no. 3: 94. https://doi.org/10.3390/axioms8030094